What does the notation $A^{-2}$ mean if $A$ is a matrix?

Question

$A^2$ means to multiple the matrix by itself, and $A^{-1}$ refers to the matrix's inverse. Would $A^{-2}$ be the square of the inverse or the inverse of the square?

Answer 1

It means $(A^{-1})^2$, so the square of the inverse, which also happens to be the inverse of the square.

Answer 2

$$\left(A^{-1}\right)^2A^2=\left(A^{-1}A^{-1}\right)(AA) = A^{-1}\left(A^{-1}A\right)A=A^{-1}IA=A^{-1}A=I $$ which says that $$\left(A^{-1}\right)^2=\left(A^2\right)^{-1} $$ This is essentially the same as the proof that $$\left({1\over x}\right)^2={1\over x^2}$$

Answer 3

It can be shown, via induction, that $(A^{-1})^n=(A^n)^{-1}$, $\forall n \in \mathbb{N}$. Thus, whenever $A$ is invertible, $A^{-n}:= (A^{-1})^n=(A^n)^{-1}$.